MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  minmar1fval Structured version   Visualization version   GIF version

Theorem minmar1fval 21776
Description: First substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
Hypotheses
Ref Expression
minmar1fval.a 𝐴 = (𝑁 Mat 𝑅)
minmar1fval.b 𝐵 = (Base‘𝐴)
minmar1fval.q 𝑄 = (𝑁 minMatR1 𝑅)
minmar1fval.o 1 = (1r𝑅)
minmar1fval.z 0 = (0g𝑅)
Assertion
Ref Expression
minmar1fval 𝑄 = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))
Distinct variable groups:   𝐵,𝑚   𝑖,𝑁,𝑗,𝑘,𝑙,𝑚   𝑅,𝑖,𝑗,𝑘,𝑙,𝑚
Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘,𝑚,𝑙)   𝐵(𝑖,𝑗,𝑘,𝑙)   𝑄(𝑖,𝑗,𝑘,𝑚,𝑙)   1 (𝑖,𝑗,𝑘,𝑚,𝑙)   0 (𝑖,𝑗,𝑘,𝑚,𝑙)

Proof of Theorem minmar1fval
Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 minmar1fval.q . 2 𝑄 = (𝑁 minMatR1 𝑅)
2 oveq12 7277 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
3 minmar1fval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
42, 3eqtr4di 2797 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
54fveq2d 6772 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
6 minmar1fval.b . . . . . 6 𝐵 = (Base‘𝐴)
75, 6eqtr4di 2797 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
8 simpl 482 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
9 fveq2 6768 . . . . . . . . . . 11 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
10 minmar1fval.o . . . . . . . . . . 11 1 = (1r𝑅)
119, 10eqtr4di 2797 . . . . . . . . . 10 (𝑟 = 𝑅 → (1r𝑟) = 1 )
12 fveq2 6768 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
13 minmar1fval.z . . . . . . . . . . 11 0 = (0g𝑅)
1412, 13eqtr4di 2797 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1511, 14ifeq12d 4485 . . . . . . . . 9 (𝑟 = 𝑅 → if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)) = if(𝑗 = 𝑙, 1 , 0 ))
1615ifeq1d 4483 . . . . . . . 8 (𝑟 = 𝑅 → if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))
1716adantl 481 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))
188, 8, 17mpoeq123dv 7341 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))
198, 8, 18mpoeq123dv 7341 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗)))) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))
207, 19mpteq12dv 5169 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗))))) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
21 df-minmar1 21765 . . . 4 minMatR1 = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗))))))
226fvexi 6782 . . . . 5 𝐵 ∈ V
2322mptex 7093 . . . 4 (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) ∈ V
2420, 21, 23ovmpoa 7419 . . 3 ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
2521mpondm0 7501 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = ∅)
26 mpt0 6571 . . . . 5 (𝑚 ∈ ∅ ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) = ∅
2725, 26eqtr4di 2797 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = (𝑚 ∈ ∅ ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
283fveq2i 6771 . . . . . . 7 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
296, 28eqtri 2767 . . . . . 6 𝐵 = (Base‘(𝑁 Mat 𝑅))
30 matbas0pc 21537 . . . . . 6 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
3129, 30eqtrid 2791 . . . . 5 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
3231mpteq1d 5173 . . . 4 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) = (𝑚 ∈ ∅ ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
3327, 32eqtr4d 2782 . . 3 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))))
3424, 33pm2.61i 182 . 2 (𝑁 minMatR1 𝑅) = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))
351, 34eqtri 2767 1 𝑄 = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1541  wcel 2109  Vcvv 3430  c0 4261  ifcif 4464  cmpt 5161  cfv 6430  (class class class)co 7268  cmpo 7270  Basecbs 16893  0gc0g 17131  1rcur 19718   Mat cmat 21535   minMatR1 cminmar1 21763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-cnex 10911  ax-1cn 10913  ax-addcl 10915
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-nn 11957  df-slot 16864  df-ndx 16876  df-base 16894  df-mat 21536  df-minmar1 21765
This theorem is referenced by:  minmar1val0  21777
  Copyright terms: Public domain W3C validator